Boundedness of the Likelihood Function in Linear Structural Equation Models

• Activity Number: 301
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Christopher Fox*+ and Andreas Käufl and Mathias Drton and Guillaume Pouliot
• Companies: The University of Chicago and University of Augsburg and The University of Chicago and The University of Chicago
• Keywords: graphical model ; structural equation model ; likelihood ; normal distribution ; covariance matrix ; mixed graph

Abstract

Linear structural equation models arise from linear relationships between random variables along with an added stochastic Gaussian noise term. These models can be represented via cyclic mixed graph models with directed edges signifying the linear dependence between variables and bidirected edges indicating possible nonzero correlations. Tian [2005] introduced a decomposition of linear structural equation models into a set of submodels that partitions the parameter space of the original model. For acyclic mixed graph models, we use this decomposition to provide an exact number of observations required to guarantee boundedness of the likelihood function over the set of possible covariance matrices with probability one. We partially extend our result to cyclic mixed graph models and provide an upper bound on the number of observations needed to ensure the likelihood function is almost surely bounded. For certain models, the number of observations required for the likelihood function to be almost surely bounded is considerably less than the number of variables.